As a preliminary remark, we observe that the
assumptions (REF ), (REF ),
() and the
estimate () in
Lemma REF  imply
\( \int _{B_{2\rho }(a_j)\setminus B_\rho (a_j)}\left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_\varepsilon  \right|^2_g)^2\right) \mathrm {vol}_g= \mathrm {o}_{\varepsilon \rightarrow 0}(\left| \log \varepsilon  \right|)\) 
We split the proof in several steps.
Step 1 (Reduction to the Euclidean setting) 
Let \(\lbrace \tau _1, \, \tau _2\rbrace \) 
be an orthonormal tangent frame on \(B_{2\rho }(a_j)\) , whose associated
connection form \(\mathcal {A}\)  satisfies (REF ).
By working in geodesic normal coordinates, we can represent
\(u_\varepsilon \)  by means of a Euclidean vector
field \(v_\varepsilon \colon \bar{B}_{2\rho }\rightarrow \mathbb {R}^2\) , as in (REF ).
Equation (REF ) implies
\(\int _{\bar{B}_{2\rho }} \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x&\stackrel{(\ref {hp:H1bounds-energyapp})}{\le }C\left| \log \varepsilon  \right|  \\\int _{\bar{B}_{2\rho }\setminus \bar{B}_\rho }\bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x&\stackrel{(\ref {ubH1-0})}{=}\mathrm {o}_{\varepsilon \rightarrow 0}(\left| \log \varepsilon  \right|)  \\\int _{\partial \bar{B}_\rho }\bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}{H}^1&\stackrel{(\ref {eq:upper_bound_bdapp})}{\le }\frac{C\left| \log \varepsilon  \right|}{\rho } \) 

where
\(\bar{e}_\varepsilon (v_\varepsilon ) := \frac{1}{2}\left| \nabla v_\varepsilon  \right|^2+ \frac{1}{4\varepsilon ^2}(1 - \left| v_\varepsilon  \right|^2)^2\) 
is the Euclidean Ginzburg-Landau energy density.
Let \(\Phi \colon B_{2\rho }(a_j)\rightarrow \bar{B}_{2\rho }\)  be the coordinate map.
The Euclidean vorticity of \(v_\varepsilon \) , i.e. the 2-form \(\mathrm {d}\bar{\jmath }(v_\varepsilon )\) 
(where \(\bar{\jmath }\)  is defined by (REF )), satisfies
\(\mathrm {d}\bar{\jmath }(v_\varepsilon ) = (\Phi ^{-1})^*\omega (u_\varepsilon ) - \mathrm {d}\mathfrak {R}_\varepsilon \qquad \textrm {where } \mathfrak {R}_\varepsilon := \left(1 - \left| v_\varepsilon  \right|^2\right) (\Phi ^{-1})^*\mathcal {A},\) 
because of Lemma REF . The energy
estimate (REF ) implies \(\mathfrak {R}_\varepsilon \rightarrow 0\) 
strongly in \(L^2(\bar{B}_{2\rho })\)  as \(\varepsilon \rightarrow 0\) ,
hence \(\mathrm {d}\mathfrak {R}_\varepsilon \rightarrow 0\)  strongly in \(W^{-1,2}(\bar{B}_{2\rho })\)  and
\( \mathrm {d}\bar{\jmath }(v_\varepsilon )\rightarrow 2\pi d_j \delta _0\qquad \textrm {in } W^{-1,1}(\bar{B}_{2\rho })\quad \textrm {as } \varepsilon \rightarrow 0.\) 

Step 2 (Construction of a suitable cover) 
From now on, we denote by \(C\)  a generic positive constant,
that may change from line to line; it may depend on \(\rho \) 
(which is fixed), but does not depend on \(\varepsilon \) . Let
\(h(\varepsilon ) := \left| \log \varepsilon  \right|^{-2}\) 
We construct a finite collection \({G}_\varepsilon \)  of Lipschitz, closed sets
that satisfies the following properties:
(i)
\(\bar{B}_{2\rho } = \bigcup _{K\in {G}_\varepsilon } K\) 
and the interiors of \(K\)  are pairwise disjoint;
(ii)
for any \(K\in {G}_\varepsilon \) , there exists
an invertible, Lipschitz map \(T_K\colon K\rightarrow [-h(\varepsilon ), \, h(\varepsilon )]^2\) ,
with Lispchitz inverse, such that
\(\left\Vert  \nabla T_K \right\Vert _{L^\infty (K)}+ \left\Vert  \nabla (T^{-1}_K) \right\Vert _{L^\infty ([-h(\varepsilon ), \, h(\varepsilon )]^2)} \le C\) 
for some constant \(C\)  that does not depend on \(\varepsilon \) , \(K\) ;
(iii)
for \(R_\varepsilon := \cup _{K\in {G}_\varepsilon }\partial K\) ,
there holds \({H}^1(R_\varepsilon ) \le Ch(\varepsilon )^{-1}\)  and
\( \int _{R_\varepsilon } \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}{H}^1\le \frac{C\left| \log \varepsilon  \right|}{h(\varepsilon )}\) 
for some \(\varepsilon \) -independent constant \(C\) ;
(iv)
for any \(K\in {G}_\varepsilon \) , either
\(\mathrm {interior}(K)\subseteq \bar{B}_{2\rho }\setminus \bar{B}_\rho \qquad \textrm {or} \qquad \mathrm {interior}(K)\subseteq \bar{B}_{\rho }.\) 

First, we define a suitable cover of \(\bar{B}_\rho \) .
Let \(T\colon \mathbb {R}^2\rightarrow \mathbb {R}^2\) , \(T(x) := \rho x/\max (\left| x^1 \right|, \, \left| x^2 \right|)\) .
The map \(T\)  is Lipschitz, invertible, has Lipschitz inverse, and maps
the square \([-\rho , \, \rho ]^2\)  to the disk \(\bar{B}_\rho \) .
Let \(N_\varepsilon \in \mathbb {N}\)  be the integer part of \(2\rho /h(\varepsilon )\) .
By Fubini theorem, we can select suitable points
\(a_\varepsilon ^0 = -\rho < a_\varepsilon ^1 < \ldots < a_\varepsilon ^{N_\varepsilon } = \rho \) 
such that, for any \(k\in \lbrace 1, \, \ldots , \, N_\varepsilon - 1\rbrace \) ,
\(\left| a_\varepsilon ^k - \left(-\rho + \frac{2\rho k}{N_\varepsilon }\right) \right|\le \frac{\rho }{2N_\varepsilon }  \\\int _{\lbrace a_\varepsilon ^k\rbrace \times [-\rho , \, \rho ]}\bar{e}_\varepsilon (v_\varepsilon \circ T) \, \mathrm {d}{H}^1\le \frac{N_\varepsilon }{\rho } \int _{[-\rho , \, \rho ]^2}\bar{e}_\varepsilon (v_\varepsilon \circ T) \, \mathrm {d}x \) 
We observe that \(2\rho /N_\varepsilon \sim h(\varepsilon )\)  as \(\varepsilon \rightarrow 0\) 
and hence, () implies
\( \int _{\lbrace a_\varepsilon ^k\rbrace \times [-\rho , \, \rho ]}\bar{e}_\varepsilon (v_\varepsilon \circ T) \, \mathrm {d}{H}^1\le \frac{C}{h(\varepsilon )} \int _{\bar{B}_\rho }\bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x\stackrel{(\ref {ubH11-1})}{\le } \frac{C\left| \log \varepsilon  \right|}{h(\varepsilon )}\) 
In fact, (REF ) remains valid for \(k=0\)  and \(k=N_\varepsilon \) ,
thanks to (). In a similar way, we find points
\(b_\varepsilon ^0 = -\rho < b_\varepsilon ^1 < \ldots < b_\varepsilon ^{N_\varepsilon } = \rho \) 
such that, for any \(h\in \lbrace 1, \, \ldots , \, N_\varepsilon - 1\rbrace \) ,
\(\left| b_\varepsilon ^h - \left(-\rho + \frac{2\rho h}{N_\varepsilon }\right) \right|\le \frac{\rho }{2N_\varepsilon }  \\\int _{[-\rho , \, \rho ]\times \lbrace b_\varepsilon ^h\rbrace }\bar{e}_\varepsilon (v_\varepsilon \circ T) \, \mathrm {d}{H}^1\le \frac{N_\varepsilon }{\rho } \int _{[-\rho , \, \rho ]^2}\bar{e}_\varepsilon (v_\varepsilon \circ T) \, \mathrm {d}x \le \frac{C\left| \log \varepsilon  \right|}{h(\varepsilon )}\) 
We consider the collection of sets \(K_{k,h} :=T([a^k_\varepsilon , \, a^{k+1}_\varepsilon ]\times [b^h_\varepsilon , \, b^{h+1}_\varepsilon ])\) , for \(k\in \lbrace 0, \, \ldots , \, N_\varepsilon - 1\rbrace \) 
and \(h\in \lbrace 0, \, \ldots , \, N_\varepsilon -1\rbrace \) . These sets cover \(\bar{B}_\rho \) ,
have pairwise disjoint interiors, satisfy the condition (ii) above
(thanks to (REF ), (REF )) and
\(\int _{\bigcup _{k,h} \partial T_{k,h}} \bar{e}_\varepsilon (v_\varepsilon ) \,\mathrm {d}{H}^1\le \frac{C\left| \log \varepsilon  \right|}{h(\varepsilon )}\) 
We construct a cover of \(\bar{B}_{2\rho }\setminus \bar{B}_\rho \) 
by considering suitable points
\(\rho _\varepsilon ^0 = \rho < \rho _\varepsilon ^1 < \ldots < \rho ^{N_\varepsilon } = 2\rho , \qquad \theta _\varepsilon ^0 = 0 < \theta _\varepsilon ^1 < \ldots < \rho ^{N_\varepsilon } = 2\pi \) 
and defining, for any \(k\in \lbrace 0, \, \ldots , \, N_\varepsilon - 1\rbrace \) ,
\(h\in \lbrace 0, \, \ldots , \, N_\varepsilon -1\rbrace \) ,
\(\tilde{K}_{k,h} := \left\lbrace (\rho \cos \theta , \, \rho \sin \theta )\colon (\rho , \, \theta )\in [\rho ^k_\varepsilon , \, \rho ^{k+1}_\varepsilon ]\times [\theta ^h_\varepsilon , \, \theta ^{h+1}_\varepsilon ]\right\rbrace \) 
For a suitable choice of \(\rho _\varepsilon ^k\)  and \(\theta _\varepsilon ^h\) 
(analogous to (REF ), ()), we can make sure that
the collection \({G}_\varepsilon := \lbrace K_{k,h}, \, \tilde{K}_{k,h}\rbrace _{k,h}\) 
satisfies all the properties (i)–(v) above.

Step 3 (Construction of a suitable measure) 
An argument based on Sobolev embeddings (e.g.,
[1]} combined with a scaling argument)
shows that
\( \sup _{x\in R_\varepsilon } \left| 1 - \left| v_\varepsilon (x) \right| \right|\le \frac{C\varepsilon ^{1/2}}{h(\varepsilon )}\left(\int _{R_\varepsilon } \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}{H}^1\right)^{1/2}\stackrel{(\ref {ubH12-1})}{\le }\frac{C\varepsilon ^{1/2}}{h(\varepsilon )^{3/2}} \left| \log \varepsilon  \right| \rightarrow 0\) 
as \(\varepsilon \rightarrow 0\) . In particular, for any \(K\in {G}_\varepsilon \) , the
projection \(v_\varepsilon /\left| v_\varepsilon  \right|\colon \partial K\rightarrow \mathbb {S}^1\) 
is well-defined and continuous. We denote by \(\deg (v_\varepsilon , \, \partial K)\) 
the topological degree of \(v_\varepsilon /\left| v_\varepsilon  \right|\)  on \(\partial K\) .
We define the measure
\( \mu _\varepsilon := 2\pi \sum _{K\in {G}_\varepsilon }\deg (v_\varepsilon , \, \partial K) \, \delta _{b_K},\) 
where \(b_K\)  is an arbitrary point in the interior of \(K\) .
Let \(\delta > 0\)  be a small parameter. Classical results by
Sandier [2]} and Jerrard [3]}
(see also [4]}
for a statement that is particular suited to our needs) imply
\( \begin{split}&\left| \deg (v_\varepsilon , \, \partial K) \right|\left(\log \frac{h(\varepsilon )}{\varepsilon }- C_\delta \left(1 + \log \left| \deg (v_\varepsilon , \, \partial K) \right|\right) \right) \\&\qquad \qquad \le C\int _K \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x + C \delta h(\varepsilon )\int _{\partial K} \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}{H}^1\end{split}\) 
for any \(K\in {G}_\varepsilon \) , where \(C_\delta \)  is a constant
that depends only on \(\delta \)  and \(\rho \) . By dividing both
sides of (REF ) by \(\left| \log \varepsilon  \right|\) , we deduce
\( \begin{split}&\left| \deg (v_\varepsilon , \, \partial K) \right|\le \frac{C}{\left| \log \varepsilon  \right|} \int _K \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x+ \frac{C\delta h(\varepsilon )}{\left| \log \varepsilon  \right|}\int _{\partial K} \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}{H}^1\end{split}\) 
By applying (REF ), () and (REF ), we deduce
\(\left| \mu _\varepsilon  \right|(\bar{B}_{2\rho })= 2 \pi \sum _{K\in {G}_\varepsilon }\left| \deg (v_\varepsilon , \, \partial K) \right| &\le C  \\\left| \mu _\varepsilon  \right|(\bar{B}_{2\rho }\setminus \bar{B}_{\rho })= 2 \pi \sum _{K\in {G}_\varepsilon \colon K\subseteq \bar{B}_{2\rho }\setminus \bar{B}_{\rho }}\left| \deg (v_\varepsilon , \, \partial K) \right| &\le C\delta \) 

Step 4 (Convergence of \(\mu _\varepsilon \) ) 
We claim that
\( \mu _\varepsilon \rightarrow 2\pi d_j \delta _0\qquad \textrm {in } W^{-1,1}(\bar{B}_{2\rho })\quad \textrm {as } \varepsilon \rightarrow 0.\) 
Let \(\varphi \in W^{1,\infty }_0(\bar{B}_{2\rho })\)  be a test function.
We have
\( \begin{split}&\left| \int _{\bar{B}_{2\rho }} \varphi \, \mathrm {d}\bar{\jmath }(v_\varepsilon )- \int _{\bar{B}_{2\rho }} \varphi \, \mathrm {d}\mu _\varepsilon  \right| \\&\qquad \le \underbrace{\left| \sum _{K\in {G}_\varepsilon }\int _{K} (\varphi - \varphi (b_K))\, \mathrm {d}\bar{\jmath }(v_\varepsilon ) \right|}_{=: I_1}+ \underbrace{\left|  \sum _{K\in {G}_\varepsilon } \varphi (b_K)\left(\int _{K} \mathrm {d}\bar{\jmath }(v_\varepsilon )- 2\pi \deg (v_\varepsilon , \, \partial K)\right) \right|}_{=:I_2}\end{split}\) 
To estimate the term \(I_1\) , we observe that
\(\left| \mathrm {d}\bar{\jmath }(v_\varepsilon ) \right| \le \left| \nabla v_\varepsilon  \right|^2\le 2\bar{e}_\varepsilon (v_\varepsilon )\)  and hence,
\( \begin{split}I_1 \le C h(\varepsilon ) \left\Vert  \nabla \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\int _{\bar{B}_{2\rho }} \bar{e}_\varepsilon (v_\varepsilon ) \, \mathrm {d}x\stackrel{(\ref {ubH11-1})}{\le } C h(\varepsilon ) \left| \log \varepsilon  \right|\left\Vert  \nabla \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\end{split}\) 
We have \(h(\varepsilon )\left| \log \varepsilon  \right| = \left| \log \varepsilon  \right|^{-1}\rightarrow 0\) 
as \(\varepsilon \rightarrow 0\) . To estimate \(I_2\) , we apply Stokes' theorem
and (REF ):
\(\begin{split}I_2 \le \left| \sum _{K\in {G}_\varepsilon }\int _{\partial K} \bar{\jmath }(v_\varepsilon )\left(1 - \frac{1}{\left| v_\varepsilon  \right|^2}\right) \right|\left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\end{split}\) 
The \(L^\infty \) -norm of \(1 - \left| v_\varepsilon  \right|^{-2}\)  on \(R_\varepsilon \) 
can be bounded from below by (REF ). Moreover, we have
\(\left| \bar{\jmath }(v_\varepsilon ) \right| \le \left| v_\varepsilon  \right| \nabla {\nabla v_\varepsilon }\) 
and, since \(\left| v_\varepsilon  \right|\)  is bounded on \(R_\varepsilon \)  by (REF ), we obtain
\(\begin{split}I_2 &\le \frac{C\varepsilon ^{1/2}\left| \log \varepsilon  \right|^{1/2}}{h(\varepsilon )^{3/2}}\left(\int _{R_\varepsilon } \bar{e}_\varepsilon (v_\varepsilon )^{1/2}\mathrm {d}{H}^1\right)\left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })} \\&\le \frac{C\varepsilon ^{1/2}\left| \log \varepsilon  \right|^{1/2}}{h(\varepsilon )^{3/2}}\left(\int _{R_\varepsilon } \bar{e}_\varepsilon (v_\varepsilon )\mathrm {d}{H}^1\right)^{1/2}{H}^1(R_\varepsilon )^{1/2} \left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\end{split}\) 
By construction, \({H}^1(R_\varepsilon ) \le C h(\varepsilon )^{-1}\) .
By applying (REF ), we deduce
\( \begin{split}I_2 \le \frac{C\varepsilon ^{1/2}}{h(\varepsilon )^{5/2}} \left| \log \varepsilon  \right|\left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}= C \varepsilon ^{1/2} \left| \log \varepsilon  \right|^6 \left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\end{split}\) 
Combining (REF ) with (REF ), (REF ),
we obtain
\(\begin{split}&\left| \int _{\bar{B}_{2\rho }} \varphi \, \mathrm {d}\bar{\jmath }(v_\varepsilon )- \int _{\bar{B}_{2\rho }} \varphi \, \mathrm {d}\mu _\varepsilon  \right| \le C\left(\left| \log \varepsilon  \right|^{-1}+ \varepsilon ^{1/2} \left| \log \varepsilon  \right|^6\right)\left(\left\Vert  \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}+ \left\Vert  \nabla \varphi  \right\Vert _{L^\infty (\bar{B}_{2\rho })}\right)\end{split}\) 
that is, \(\mathrm {d}\bar{\jmath }(v_\varepsilon ) - \mu _\varepsilon \rightarrow 0\)  strongly
in \(W^{-1,1}(\bar{B}_{2\rho })\) . Now (REF )
follows, due to (REF ).

Step 5 (Conclusion) 
Let \(\psi \in C^\infty _{\mathrm {c}}(\bar{B}_{2\rho }\)  be such that
\(0 \le \psi \le 1\)  and \(\psi = 1\)  in \(\bar{B}_\rho \) . By
construction of the measure \(\mu _\varepsilon \)  (see (REF ))
and additivity of the degree, we have \(2\pi \deg (v_\varepsilon , \, \partial \bar{B}_\rho )= \mu _\varepsilon (\bar{B}_\rho )\) . Therefore,
\( \begin{split}2\pi \deg (v_\varepsilon , \, \partial \bar{B}_{\rho })= \mu _\varepsilon (\bar{B}_{\rho }) = \int _{\bar{B}_{2\rho }} \psi \,\mathrm {d}\mu _\varepsilon - \int _{\bar{B}_{2\rho }\setminus \bar{B}_\rho } \psi \,\mathrm {d}\mu _\varepsilon \end{split}\) 
By (REF ), we have
\( \begin{split}\int _{\bar{B}_{2\rho }} \psi \,\mathrm {d}\mu _\varepsilon \rightarrow 2\pi d_j \psi (0)= 2\pi d_j\qquad \textrm {as } \varepsilon \rightarrow 0.\end{split}\) 
On the other hand,
\( \left| \int _{\bar{B}_{2\rho }\setminus \bar{B}_\rho } \psi \,\mathrm {d}\mu _\varepsilon  \right|\le \left| \mu _\varepsilon  \right| (\bar{B}_{2\rho }\setminus \bar{B}_\rho )\stackrel{(\ref {ubH13-6})}{\le } C\delta \) 
Therefore, for \(\varepsilon \)  sufficiently large we have
\(\left| \deg (v_\varepsilon , \, \partial \bar{B}_\rho ) - d_j \right| \le C\delta \) 
We choose \(\delta \)  small enough, so that \(C\delta < 1\) .
Since both \(\deg (v_\varepsilon , \, \partial \bar{B}_\rho )\)  and \(d_j\) 
are integers, we conclude that \(\deg (v_\varepsilon , \, \partial \bar{B}_\rho ) = d_j\) 
for \(\varepsilon \)  small enough, and the lemma follows.

